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PROBE–A multicriteria decision support system for portfolio robustness evaluation
ISSN 2041-4668 (Online) PROBE – A multicriteria decision support system for portfolio robustness evaluation Joóo Carlos Lourenỗo1 and Carlos A Bana e Costa1,2 CEG-IST, Centre for Management Studies of IST, Technical University of Lisbon Operational Research Group, Department of Management, London School of Economics joao.lourenco@ist.utl.pt c.bana@lse.ac.uk Working Paper LSEOR 09.108 ISSN 2041-4668 (Online) First published in Great Britain in 2009 by the Operational Research Group, Department of Management London School of Economics and Political Science Copyright © The London School of Economics and Political Science, 2009 The contributors have asserted their moral rights All rights reserved No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, without the prior permission in writing of the publisher, nor be circulated in any form of binding or cover other than that in which it is published Typeset, printed and bound by: The London School of Economics and Political Science Houghton Street London WC2A 2AE Working Paper No: LSEOR 09.108 ISSN 2041-4668 (Online) PROBE – A multicriteria decision support system for portfolio robustness evaluation Joóo Carlos Lourenỗo1 and Carlos A Bana e Costa1,2 CEG-IST, Centre for Management Studies of IST, Technical University of Lisbon Operational Research Group, Department of Management, London School of Economics joao.lourenco@ist.utl.pt c.bana@lse.ac.uk Abstract: This paper deals with the problem of selecting a robust portfolio of projects in the presence of limited resources, multiple criteria, different project interactions and several types of uncertainty We present a new decision support system, PROBE – Portfolio Robustness Evaluation, and the algorithms it implements PROBE identifies all efficient portfolios, either convex or non-convex efficient, depicting the respective Pareto frontier, within a given portfolio cost range, and allows performing indepth interactive analysis of the robustness of selecting a proposed portfolio Keywords: Portfolio decision analysis, resource allocation, portfolio robustness, DSS Working Paper LSEOR 09.108 Introduction Suppose that a manager has several projects, each one expected to add value to his organization The manager has associated a forward cost for each project and realized that there is not enough money to fund them all Therefore, he wants to select a portfolio of projects providing the organization with the best value for money An exhaustive analysis of all possible portfolios, from the empty portfolio (in which no projects are funded and no benefits are realized) to the full portfolio (that would require all projects to be funded), would be impractical even if the number of projects is not too big For example, 20 projects will give more than one million portfolios (precisely 20 = 1,048,576) A practical selection strategy would be to prioritize the projects in decreasing order of their benefit-to-cost ratios and proceeding down the list until the available budget is exhausted (Lorie and Savage 1955, Edwards 1977, Kirkwood 1997, Kleinmuntz and Kleinmuntz 1999, Bana e Costa et al 2006, Buede and Bresnick 2007, Phillips and Bana e Costa 2007) The portfolio selected by this approach produces the highest benefit for the money spent, but may not be the one that produces the maximum benefit for the money available Alternatively, the optimization approach could be followed It consists in searching for the portfolio with the highest benefit for the budget available, by solving a (knapsack) mathematical programming problem (Martello and Toth 1990, Kellerer et al 2004) that maximizes cumulative benefit without exceeding the budget constraint (Weingartner 1963, Golabi et al 1981, Kirkwood 1997, Kleinmuntz 2007) As shown in Section 2, the portfolio selected by each one of these two approaches, for the same fixed budget, is not necessarily the same, because the portfolio selected by the prioritization approach cannot include any project with a lower benefit-to-cost ratio (that is, a less “productive” or “profitable” project - Cooper et al 1999, Brealey and Myers 2003) than a non-selected project This is not necessarily the case of the portfolio selected by the optimization approach Arguments favoring each one of the two approaches are discussed in Section Resource allocation decisions often require project portfolio selections involving multiple quantitative and qualitative benefit dimensions (or criteria) In a previous paper (Lourenỗo et al 2008), we studied commercial off-the-shelf software for multicriteria portfolio analysis based on additive aggregation of multiple benefit criteria: Equity (Catalyze 2008), HiPriority (Krysalis 2007), Logical Decisions Portfolio (Logical Decisions 2008) and Expert Choice Resource Aligner (Expert Choice 2007) Equity and HiPriority follow the prioritization approach, Expert Choice Resource Aligner (Expert Choice 2007) follows the optimization approach, and Logical Decisions Portfolio implements the two approaches Section introduces PROBE – Portfolio Robustness Evaluation, a new decision support system for multicriteria portfolio analysis that implements the optimization approach but also finds the solutions given by the prioritization approach When several benefit criteria are defined, PROBE calculates the benefit value of each project by an additive value model Therefore, the basic project inputs for a multicriteria portfolio analysis are the cost of each project and its value scores on each one of the benefit criteria, and the weights that capture trade-offs between criteria (see Section 3.1) Different types of project interactions or interdependencies can also be inputted (see Section 3.2) For a resource allocation model defined with the data inputted by the user, PROBE is able to identify all efficient portfolios, distinguishing the convex from the non-convex efficient ones, and depicts the respective Pareto frontier, running the algorithms presented in Section 4.1 Several sources of uncertainty are often present in real-world resource allocation contexts, affecting the precision of some of the inputs Many times, a “best” portfolio is selected on the basis of “best guess” input data only It would then be wise to analyze the robustness of that “best” portfolio taking into account, simultaneously, uncertain data affecting the costs and benefits of the portfolios, that is, imprecise project costs and benefit values and criteria weights Portfolio robustness evaluation (see Section 4.2) has been the core motivation for the conception of PROBE, because our previous study of commercial packages (Lourenỗo et al 2008) revealed that none of them addresses robustness analysis The example in Section illustrates how a robustness analysis can be conducted with the DSS PROBE within a given uncertainty domain Some final remarks about our approach are presented in Section 6, namely a short comparison with the pioneer research of Liesiö et al (2007, 2008) on “robust portfolio modeling (RPM)” Basic concepts and portfolio selection approaches In this paper, portfolio selection is only concerned with a set X of m projects that worth funding, therefore assuming that project proposals that not worth funding were screened out in a previous phase of the selection process Conceptually, the benefit value of a project that adds no value to a portfolio should be zero; consequently, the benefit value of a project that worth funding should be defined as the value that it adds to the portfolio Let cj > and vj > be, respectively, the cost and the benefit value of project j of X (j = 1,…, m) and B the budget available For simplicity, without loss of generality, we assume that the m projects of X are presented in decreasing order by their benefit-to-cost ratio r j = vj/cj (j = 1,…, m) such that r p ≥ r p+1, p = 1,…, m-1, as the four projects (1, 2, and 4) in Table are Table Benefit values, costs and benefit-to-cost ratios of four projects Projects j vj cj 10 vj/cj 0.75 0.50 0.30 0.25 Let vij be the value score of project j on the benefit criterion i, i = 1,…, n ( n 1) and wi ( wi ) the weight of criterion i, i = 1,…, n (with n i 1 wi 1) The benefit value vj of project j is given by vj i 1 wi vij n (1) A portfolio (p) is a subset of projects of X ( p X ) and the number of possible portfolios is 2m Let cp and vp be the cost and the benefit of portfolio p given by, respectively c p j p c j (2) and v p j p vj jp i 1 wi vij n (3) A portfolio p dominates another portfolio d if cp ≤ cd and vp > vd, or if cp < cd and vp ≥ vd A portfolio is efficient (Pareto-efficient or Pareto-optimal or non-dominated) when no other portfolio dominates it Figure shows all (16=24) portfolios that can be formed with the four projects of Table (including the empty portfolio {} ) {1, 2, 3, 4} 12 {1, 2, 3} Cumulative benefit 10 {1, 2, 4} {1, 2} {2} {1} {} 0 10 12 14 16 18 20 22 24 26 28 30 Cumulative cost Figure Chart showing the portfolios that can be formed with the four projects Note Efficient portfolios are represented by triangular dots and dominated portfolios by squared dots The efficient portfolios are shown as triangular dots and form the efficient frontier (or Pareto-front), associated for simplicity with the dotted line in Figure 1, whereas the dashed line links only convex efficient portfolios (the convex efficient frontier, which is the convex hull of the Pareto-front) As can be seen in Figure 1, there are two efficient portfolios that are non-convex efficient (those that belong to the dotted line but not to the dashed line): portfolio {2} and portfolio {1, 2, 4} Given a fixed budget B, the prioritization approach selects the convex efficient portfolio formed by the projects j, j = 1,…, k with k ≤ m, such that k j 1 c j B and k 1 j 1 c j B In this approach, the notion of “value-for-money” of a project (Bana e Costa et al 2006, Phillips and Bana e Costa 2007, Phillips 2007) or its “bang-for-thebuck” (Cooper et al 2001, Brealey and Myers 2003, Buede and Bresnick 2007) is associated with the slope of the benefit-to-cost triangle of each project, as shown in Figure for the four projects of Table The last column of Table shows that the order of selection by prioritization would be: first project 1, then project 2, followed by project and finally project When the budget increases from to 30 (see Figure 2), the sequence of selected portfolios, from the empty portfolio {} to the full portfolio {1, 2,3, 4} , starts by portfolio {1} for ≤ B < 12, followed by portfolio {1, 2} for 12 ≤ B < 22, and then portfolio {1, 2,3} for 22 ≤ B < 30 {1, 2, 3, 4} 12 {1, 2, 3} Cumulative benefit 10 {1, 2} value-for-money slopes {1} {} 0 10 12 14 16 18 20 22 24 26 28 30 Cumulative cost Figure Cumulative cost versus cumulative benefit chart showing the portfolios formed by the benefit-to-cost ratio approach Notes The value-for-money of each project is given by the slope of its benefit-to-cost triangle The arrow in the value-for-money slopes box shows the direction of improvement of the benefit-to-cost value of the projects The composition of each portfolio is shown in brackets next to the corresponding dot Alternatively, the portfolio selected by the optimization approach is the optimal solution of the following mathematical programming problem (known as the knapsack problem): m maximize v x , j 1 j j m subject to: c x j 1 j j B, (4) x j 0,1 , j 1, , m, where xj is a binary variable such that xj = if project j integrates the optimal portfolio and xj = otherwise For the four projects in Table and a budget B = 20, the optimal portfolio is {1, 2, 4} with a benefit of for a cost of 20 (see Figure 1); whereas the portfolio selected by the prioritization approach, for the same budget, would be {1, 2} with a benefit of for a cost of 12 (see Figure 2) Figure shows that both portfolios are efficient, but it seems that optimization identifies a “better” portfolio than prioritization This is true for a manager only concerned with getting more benefit (9 > 7), but not necessarily if the “bang-for-the-buck” received also matters (9/20 < 7/12) Note that portfolio {1, 2, 4} includes project 4, which is “less productive” than the non-selected project (because r < r – see Table 1) Kirkwood (1997, Chapter 8.1) and Kleinmuntz (2007) also briefly discuss pros and cons of these two approaches Any efficient portfolio can be selected by the optimization approach, whereas the prioritization approach always selects a convex efficient portfolio When the optimal solution of the knapsack problem (4) is a convex efficient portfolio, the portfolio selected by prioritization is the same; but, when the optimal solution of problem (4) is a non-convex efficient portfolio, the portfolio selected by prioritization is the first convex efficient portfolio at its left in the convex efficient frontier This portfolio could be found by constraining the knapsack problem in such a way that the optimal solution does not include any project with a lower benefit-to-cost ratio than a non-selected project.1 As proved by Dantzig (1957), a portfolio formed through the prioritization approach is included in the optimal solution of a (relaxed) knapsack problem in which the projects are assumed to be divisible (see also Martello and Toth 1979) The benefit-to-cost ratio algorithm is also designated by the “greedy algorithm for the knapsack problem” in (Kellerer et al 2004, Korte and Vygen 2006) Finally, if there are at least two projects with the same benefit-to-cost ratio, some convex efficient portfolios are not identified by the prioritization approach However, the prioritization approach can easily deal with a large number of projects, contrary to knapsack optimization algorithms Indeed, the knapsack problem (4) is technically difficult to solve despite its straightforward structure, due to the integrality constraints xj {0,1} , j = 1,…, m.2 Introducing the DSS PROBE 3.1 The MCDA and PDA components and basic input data PROBE is a multicriteria decision support system for portfolio robustness evaluation that integrates two main architectural components: a multicriteria decision analysis (MCDA) component and a portfolio decision analysis (PDA) component The MCDA component allows the user to structure the benefit criteria in the form of a value tree and input data for the costs of the projects and their benefit scores on each bottomlevel criterion of the value tree Let X be a specific set of projects j (j = 1,…, m) defined by the user Even when uncertainty is present, PROBE always asks the user to input, for each project j, a (“best guess”) cost cj and (“best guess”) benefit value scores vij on each bottomlevel criterion i, i = 1,…, n (n=1 if only one benefit dimension, such as NPV, is defined) For a value tree with only one level of n>1 benefit criteria i (i = 1,…, n), (“best guess”) weights wi (i = 1,…, n) should be introduced and PROBE computes the benefit value vj of each project j (j = 1,…, m) by applying the non-hierarchical additive model (1) If the value tree has two or more levels below the root node, crisp weights are defined for the criteria at each level and PROBE uses a hierarchical value model to compute an aggregate benefit value vj of each project j (j = 1,…, m) by applying model (1) bottom-up successively If a branch of risk criteria is included in the value tree set of criteria, the vj of each project j is more adequately designated by “risk-adjusted benefit” (Phillips and Bana e Costa 2007) For the sake of simplicity, without loss of generality, all programs and algorithms presented in this paper assumes a non-hierarchical benefit model, which can be easily extended to the corresponding generic hierarchical formulation implemented in PROBE The knapsack problem is considered to be a nondeterministic polynomial-time hard (NP-hard) problem (for a discussion on complexity and hardness see Garey and Johnson 1979) A significant number of exact and approximate resolution algorithms for this problem have been thoroughly studied (Martello and Toth 1990, Kellerer et al 2004) For the given project costs cj and benefit scores vj (j = 1,…, m), the PDA component solves the knapsack optimization problem (4) – with or without additional linear constraints possibly added by the user to model project interactions (see Section 3.2) – for any fixed budget B, finds all efficient portfolios, distinguishes convex from non-convex ones (see Section 4.1) – a functionality not included in the software packages analyzed in (Lourenỗo et al 2008) – and displays the portion of the Pareto-front for an user-defined limited portfolio cost range [ B, B] (see section 4.1) When the number of projects of X is compatible with a reasonable computational time (see the Appendix), PROBE automatically displays the full efficient frontier, assuming by default B = and B j 1 c j m Concerning the modeling of uncertainty, PROBE allows the user to input: a set c of plausible cost ranges [ c j , c j ] such that cj c j c j (j = 1,…, m); a set v of plausible benefit scores ranges [ vij , vij ] such that vij vij vij (i = 1,…, n; j = 1,…, m); linear relationships on the weights (for example, weights rankings and/or weights ranges) defining a set w of feasible weights such that wi w (i = 1,…, n) The MCDA component uses the additive model to calculate by optimization the feasible benefit value range [ vj , v j ] defined by v w for each project j (j = 1,…, m) as follows: vj i 1 wi vij and v j max i 1 wi vij It n w n w is within a user-defined uncertainty domain = c v w that portfolio robustness evaluation takes place (see Section 4.2) PROBE is coded in the C++ programming language (Stroustrup 1994, 1997) using the C++ Builder development software having Microsoft Windows as its target environment PROBE uses the mixed integer linear programming (MILP) solver “lp_solve” 5.5.0.14 (available at http://sourceforge.net/projects/lpsolve/) to solve the optimization problems included in the PROBE algorithms presented in Sections 4.1 and 4.2 This solver is based on the revised simplex method and on the branch-and-bound algorithm to deal with integer variables Mutual inclusivity of two projects i and j xi xj (10) Note: Dependencies or mutual inclusivity among more than two projects are easily modeled by adding extra constraints of types (9) or (10), respectively Mutual exclusivity of two projects i and j xi xj (11) Mutual exclusivity of more than two projects are easily modeled by adding extra binary variables in the left hand side of (11) Group constraints on a subset G of nG projects (1 ≤ nG ≤ n), such as (with ≤ q ≤ nG) jG x j q, (12) jG x j q, (13) jG xj q (14) PROBE includes an interface that allows the user to add to problem (4) any other type of linear constraint A feasible portfolio is a portfolio that respects all of the constraints introduced by the user PROBE innovative functionalities 4.1 Finding all efficient portfolios within a given portfolio cost range For project costs cj and benefit scores vj (j = 1,…, m) given by the MCDA component, and supposing for now that no project interaction constraints were defined, the PDA component starts searching for the efficient portfolios, within a given portfolio cost range [ B, B] , by solving problem (4) with B B Next, problem (4) is again solved with B equal to the cost of the optimal portfolio previously found minus a small enough amount ; and so on while B B The algorithm designed to implement this process, FindEfficientPortfolios, presented in Figure 3, is also capable of identifying all possible multiple optimal solutions Finally, PROBE uses another algorithm, FindConvexEfficientPortfolios, presented in Figure 4, to differentiate convex efficient from non-convex efficient portfolios Additional linear constraints of the types described in Section can easily be added to both algorithms to take project interactions into account when finding efficient portfolios Algorithm FindEfficientPortfolios(c, v, B , B ) Given the cost cj and benefit value vj of each project j (j= 1, ,m) of a set X of m projects, algorithm FindEfficientPortfolios finds the whole set of efficient portfolios for a given portfolio cost range, defined by its lower and upper bounds B and B , respectively The optimization problem inside each solve call is presented including only the cost constraint, but other linear constraints can easily be added The efficient portfolios found by the algorithm are stored in a matrix mEP with a number of rows equal to the number of efficient portfolios and a number of columns equal to m+ Each row stores the data of one efficient portfolio, with its cost inserted in the first cell and its benefit value in the second one, and each of the other m cells corresponding to each project j, j=1,…m, in such a way that is inserted in cell j+2 if project j is included in the portfolio, or otherwise When searching for multiple optimal portfolios with the same cost, each time an optimal portfolio is found a constraint is added to the optimization problem to prevent that portfolio to be found again; the set of these constraints is designated by NRS At the end, the efficient portfolios stored in matrix mEP are sorted by increasing order of cost initialization B : B ; stop := false; duplicate := false; := 10-6; r := search for efficient portfolios while (stop = false) if (duplicate = false) then m solve ( z : max j 1 vj xj , s.t else m j 1 c j xj B , xj 0,1 ) search for other optimal portfolios with the same cost m solve ( z : max j 1 vj xj , s.t j 1 c j xj B , NRS, xj 0,1 ) m end if if (a portfolio is found and m j 1 c j x*j B ) then add one row to mEP; r := r + mEP[r][1] := m j 1 c j x*j mEP[r][2] := z for (j := to m) mEP[r][j + 2] := x*j end for if (duplicate = false) then stores the cost of the portfolio found stores the benefit of the portfolio found stores the solution found B : j 1 c j x*j ; duplicate := true m end if for (j := to m) prepares the LHS of a new NRS constraint sets the variables with the solution found xj : x*j end for m m add to the optimization problem the new NRS constraint j 1 xj j 1 x*j else if (duplicate = true) remove the NRS constraints from the optimization problem B : j 1 c j x*j ; duplicate := false m else stop := true end if end while reverse the order of portfolios in mEP return mEP to get them sorted by increasing cost Figure Algorithm FindEfficientPortfolios 10 Algorithm FindConvexEfficientPortfolios(mEP) Given a matrix mEP of efficient portfolios found with algorithm FindEfficientPortfolios, algorithm FindConvexEfficientPortfolios finds which ones are convex efficient A new column denoted as “lastCol” is added to mEP to register which portfolios (one by each row) is, or it is not, convex efficient, in such a way that is inserted in cell lastCol if the corresponding portfolio is convex efficient or otherwise initialization stop := false; pos := 1; r := 1; lastCol:= m+3 endPos := number of rows of the matrix mEP add one more column to the right of mEP and set all the values of the lastCol to mEP [1][j] := records the first entry of mEP as the first efficient portfolio if there is only one efficient portfolio the algorithm stops if (endPos = 1) then exit end if search for convex efficient portfolios while (stop = false) maxSlope := -1 maxIter := -1 for (iter := pos + to endPos) if (mEP [iter][2] - mEP [pos][2] = 0) then maxIter := iter exit for else slope := (mEP [iter][2] - mEP [pos][2])/( mEP [iter][1] - mEP [pos][1]) if (slope > maxSlope) then maxSlope := slope; maxIter := iter end if end if end for mEP [bestIter][lastCol] := records the efficient portfolio found if (maxIter = endPos) then stop := true else r := r +1; pos := maxIter end if end while return mEP Figure Algorithm FindConvexEfficientPortfolios 11 4.2 Portfolio robustness evaluation For the given costs cj and benefit scores vj (j = 1,…, m) of the projects, let p*, with cost cp* and benefit vp*, be a specific efficient portfolio selected by the user, either by asking PROBE to found the optimal solution of problem (4) for a fixed budget B or by inspection of the efficient portfolios found by PROBE within a portfolio cost range [ B, B] In one situation or the other, the user may be concerned with the robustness of the choice of p* in face of an uncertainty domain (see Section 3.1), within which c p* c p* c p* and v p* v p* v p* Portfolio robustness evaluation (PROBE) is based on the extension of the MCDA notion of “additive dominance” between projects (Bana e Costa 1990, Bana e Costa and Vincke 1995) implemented in the original preference robustness evaluation PROBE software (Bana e Costa 2001, Bana e Costa et al 2007) Given an uncertainty domain , we say that portfolio p* additively dominates portfolio p if and only if the result z of optimization problem (15) is non-negative: z : wi vij vij vij j p i 1 j p j p* j p j p * * n subject to: c c j p j p* j j p* j p j 0, (15) wi w , i 1,, n Which portfolios p (henceforth called candidates) should be pairwise compared with portfolio p*? Using algorithm FindCandidates, described in Figure 5, PROBE takes as candidate any portfolio p such that there exists at least a point in the given uncertainty domain where the cost of p is not higher than the cost of p* and the benefit of p is not smaller than the benefit of p* Next, PROBE uses algorithm FindNonDominatedPortfolios, described in Figure 6, to solve problem (15) for each candidate portfolio p and find which portfolios p (the competitor portfolios) are not additively dominated by p* (the proposed portfolio) 12 Algorithm FindCandidates(, p*) Algorithm FindCandidates selects “candidate portfolios” p that may be non-dominated by p* given an uncertainty domain The candidate portfolios p are stored in matrix mCP initialization c p* : j p* c j stores the upper bound cost of the efficient portfolio p* compute v p* store vj* , j p* computes the minimum benefit value of the efficient portfolio p* given v w vS : M stop := false M denotes a huge positive finite real number the v j * are the benefit values of projects j p * when the benefit value of p* is minimum search for candidate portfolios while (stop = false) solve( z : max vj xj j p * v x , s.t c x c x j p * j j j p * j j j p* j j c p* , xj 0,1 , j 1,, m ) if ( z v p* ) then add one row to mCP store the result z and the portfolio found p in the new added row of mCP if ( z vS ) then vS : z if (there are bound or/and NRS constraints added on previous iterations) then remove the bound and NRS constraints previously added endif add to the optimization problem the bound constraint for (j := to m) xj : xj m j 1 v j xj vS prepares the LHS of a new NRS constraint sets the variables to the solution found end for add to the optimization problem a new NRS constraint m j 1 xj j 1 xj m else for (j := to m) xj : xj prepares the LHS of a new NRS constraint sets the variables with the solution found end for add to the optimization problem a new NRS constraint m j 1 xj j 1 xj endif else stop := true endif end while return mCP Figure Algorithm FindCandidatePortfolios 13 m Algorithm FindNonDominatedPortfolios(v, w, p*, mCP) Algorithm FindNonDominatedPortfolios searches for portfolios that are non-dominated (the competitor portfolios) by the efficient portfolio p*(the proposed portfolio) given an uncertainty domain The algorithm analyzes the matrix mCP of candidate portfolios p previously found by algorithm FindCandidates The non-dominated portfolios found by algorithm FindNonDominatedPortfolios are stored in a matrix mNDP initialization k := search for non-dominated portfolios for (k := to the number of rows of mCP) p := mCP [k] the projects of the k candidate portfolio are used to define p solve ( z : wi vij vij vij , s.t wi w , i 1,, n ) i 1 j p j p j p * j p * j p * if the portfolio p is non-dominated by p* if ( z ) then add one row to mNDP store the result z and the portfolio p in the new added row of mNDP k := k + end if end for return mNDP n Figure Algorithm FindNonDominatedPortfolios We say that the choice of p* is undoubtedly robust when no portfolio p exists Otherwise, for each portfolio p PROBE solves problem (16) z : max wi vij vij vij j p i 1 j p j p* j p * j p * n wi w , i 1,, n subject to: (16) The result z is the upper bound of the range of variation of the difference between the benefit value of the proposed portfolio p* and the benefit value of the competitor portfolio p in the uncertainty domain : v p* v p z, z , with the lower bound z already calculated by problem (15) Finally, using expressions (17) and (18), PROBE calculates the lower bound d and upper bound d of the range of variation of the difference between the costs of p* and p in : c p* c p d , d d c c j (17) c c j (18) j p j p* d j p j p* j j j p* j p j p* j p 14 The user is then in conditions of analyzing the differences in cost and benefit between the proposed portfolio p* and each competitor portfolio p , as illustrated with an example in Section Finally, “star projects” can be identified as those projects that are common to the proposed portfolio p* and all of its competitors Example A manager has to allocate resources to 12 projects from four departments, totaling €14 million (see Table 2), largely exceeding the €5 million available The manager and the four department directors constitute the decision making group (DMG) that developed a portfolio decision analysis with the support of PROBE The DMG evaluated the added value of each project on each one of four benefit criteria (B1, B2, B3 and B4) The respective benefit value scores of the projects are shown in columns “B1” to “B4” in Table Using this data and the criteria weights indicated in the last row of Table 2, the MCDA component of PROBE computed the benefit values of the projects shown in the last column of Table Table Basic input data of the 12 projects and MCDA output Project P01 P02 P03 P04 P05 P06 P07 P08 P09 P10 P11 P12 Weights Dept A A A A B B B B C C D D Cost (€106) 1.1 1.9 0.9 0.9 1.8 1.3 1.3 1.1 0.9 1.3 0.7 0.8 =14 B1 67 55 100 90 48 43 42 40 80 88 41 86 0.31 Benefit value scores B2 B3 40 30 37 40 90 80 80 70 35 40 32 25 64 80 50 44 90 78 66 80 48 49 88 90 0.19 0.29 B4 50 20 90 95 33 44 55 20 66 70 25 72 Benefit value 47.57 39.88 90.20 83.35 40.06 35.90 59.93 38.86 78.38 77.72 41.29 84.60 0.21 Figure is a snapshot of the main window PROBE, displayed by the PDA component for the basic input data in Table The portfolio selected by the optimization approach, solving the knapsack problem (4) for a budget of €5 million, is the efficient portfolio (4.8, 414.25)3, indicated by a star dot in the graph, with cost €4.8 million and a benefit of 414.25 and composed by five projects, P03 and P04 of Dept A, P09 and P10 of Dept C and P12 from Dept D, as highlighted in the tables above the graph For the sake of brevity, we will use the ordered pair (x, y) to denote the portfolio with cost €x million and benefit y 15 Figure PROBE analysis of efficient portfolios Notes Each row of the top-left table shows information about one efficient portfolio: a or a in column “CE” indicates whether the portfolio is convex efficient or non-convex efficient, respectively; a or a in each one of the columns“P01” to “P12” indicate whether the respective project is included or not, respectively, in the portfolio The top-right table window shows the projects structured within areas (the four departments) Each efficient portfolio is represented by a dot in the bottom graph (where the star dot indicates the selected portfolio), with the convex efficient ones linked by a dotted-line This optimal portfolio (4.8, 414.25) is convex efficient, therefore it is also the one selected by the prioritization approach as shown in Table (This implication would not be necessarily true if additional project interaction constraints had been defined.) Table Projects ranked by decreasing benefit-to-cost ratio (order of priority) Project P12 P03 P04 P09 P10 P11 P07 P01 P08 P06 P05 P02 Benefit 84.6 90.2 83.35 78.38 77.72 41.29 59.93 47.57 38.86 35.9 40.06 39.88 Cost 0.8 0.9 0.9 0.9 1.3 0.7 1.3 1.1 1.1 1.3 1.8 1.9 Ratio 105.8 100.2 92.6 87.1 59.8 59.0 46.1 43.2 35.3 27.6 22.3 21.0 16 Cumulative cost 0.8 1.7 2.6 3.5 4.8 5.5 6.8 7.9 9.0 10.3 12.1 14.0 The director of Dept B argued against the selection of portfolio (4.8, 414.25) because it does not include any project from his department The DMG decided to analyze what would be the loss of benefit of imposing that at least one project from each department must be selected Accordingly, four constraints of type (13) with q = were added to PROBE, giving rise to the new results shown in Figure Figure PROBE analysis of efficient portfolios with group constraints The optimal portfolio is now (4.8, 396.46), by coincidence with the same cost of portfolio (4.8, 414.25) but giving less 17.79 units of benefit, caused by replacing P10 by P07, as well noted by the director of Dept C The robustness of selecting portfolio (4.8, 396.46) was then evaluated for an uncertainty domain defined by all benefit value scores of all projects on all criteria varying ±10 units and the weights varying within the bounds indicated in Table Table Uncertainty on the weights Criterion Weight B1 B2 B3 B4 0.31 0.19 0.29 0.21 Lower bound 0.26 0.14 0.24 0.16 17 Upper bound 0.36 0.24 0.34 0.26 ... approach, and Logical Decisions Portfolio implements the two approaches Section introduces PROBE – Portfolio Robustness Evaluation, a new decision support system for multicriteria portfolio analysis... multicriteria decision support system for portfolio robustness evaluation that integrates two main architectural components: a multicriteria decision analysis (MCDA) component and a portfolio decision analysis... selected portfolios, from the empty portfolio {} to the full portfolio {1, 2,3, 4} , starts by portfolio {1} for ≤ B < 12, followed by portfolio {1, 2} for 12 ≤ B < 22, and then portfolio {1, 2,3} for